Variable-order Mittag-Leffler fractional operator and application to mobile-immobile advection-dispersion model

The main target of our research is to achieve the solution to a variable-order fractional mobile-immobile advection-dispersion equation within the Atangana-Baleanu (AB) fractional operator. This equation is the best tool for modeling dynamical systems. The presented scheme is based on the collocation method and Lagrange polynomials. The given approach uses derivative operational matrices of integer and variable orders. The operational matrix of variable-order derivative in the AB sense is computed based on Lagrange polynomials in the presented study. The key theory of this approach is to convert the corresponding problem to a system of algebraic equations. The solution to the problem under study can be computed by utilizing the mentioned system and the collocation points. The aim of this paper is to show that this technique is easily used to solve fractional partial differential equations with excellent accuracy in results. To demonstrate the accuracy and efficiency of the suggested method, some numerical examples are provided.